And with six goals to his name, the 25-year-old has now scored double the amount as his nearest competitors, including Ronaldo.
“Antoine Griezmann will unquestionably be the player of the tournament, unless Ronaldo scores a hat-trick in the final,” wrote Lineker in his Mail on Sunday column.
“He's been a breath of fresh air. He's finished beautifully, his movement is great, as is his touch and he plays delightful football.
“He's diminutive, with that angel-like face. And he's enjoying it. But then you do when it's going well - it's easy to enjoy when the goals are going in and the French team are really enjoying the atmosphere in the stadiums and cities where they play.”
While Griezmann has led the charge in attack, Manchester United-target Paul Pogba has pulled the strings in midfield for France.
However, Lineker believes the 23-year-old has some way to go to match other French legends such as Zinedine Zidane.
“Paul Pogba is still very young at 23. He's far from a Zinedine Zidane or a Michel Platini at present,” he wrote.
“He could turn into that. You can see there's a lot of ability there. He covers the ground well, his technique is very good and he's a threat in the air.
“It's whether he takes the next step up to be a great player. He clearly has bags of potential.”